A228097 Integer areas of excentral triangles of integer-sided triangles.

30, 50, 75, 120, 195, 200, 260, 270, 300, 340, 450, 480, 510, 525, 585, 675, 700, 750, 765, 780, 800, 845, 1014, 1015, 1040, 1080, 1156, 1200, 1250, 1360, 1365, 1470, 1530, 1554, 1740, 1755, 1800, 1845, 1875, 1920, 2040, 2100, 2210, 2220, 2275, 2340, 2430

Offset

1,1

Comments

The excentral triangle, also called the tritangent triangle, of a triangle ABC is the triangle IJK with vertices corresponding to the excenters of ABC.

The excentral triangle has side lengths:

a' = a*csc(A/2) where csc(z)=1/sin(z);

b' = b*csc(B/2);

c' = c*csc(C/2);

and area:

A' = 4*A*a*b*c/((a+b-c)*(a-b+c)*(-a+b+c)).

Property of this sequence:

The areas of the original triangles are integers. The primitive triangles with areas a(n) are 30, 50, 75, 195, ...

The non-primitive triangles with areas 4*a(n) are in the sequence.

The following table gives the first values (A', A, a, b, c) where A' is the area of the excentral triangles, A is the area of the reference triangles ABC, a, b, c the integer sides of the original triangles ABC.

----------------------

| A'| A | a| b| c|

----------------------

| 30| 6| 3| 4| 5|

| 50| 12| 5| 5| 6|

| 75| 12| 5| 5| 8|

|120| 24| 6| 8| 10|

|195| 30| 5| 12| 13|

|200| 48| 10| 10| 12|

|260| 24| 4| 13| 15|

|270| 54| 9| 12| 15|

|300| 48| 10| 10| 16|

|340| 60| 8| 15| 17|

......................

References

C. Kimberling, Triangle Centers and Central Triangles. Congr. Numer. 129, 1-295, 1998.

Links

Table of n, a(n) for n=1..47.

Wolfram MathWorld, Excentral Triangles

Example

30 is in the sequence because the area A' = 4*A*a*b*c/((a+b-c)*(a-b+c)*(-a+b+c)) of the excentral triangle corresponding to the initial triangle (3,4,5) is A' = 4*6*3*4*5/((3+4-5)*(3-4+5)*(-3+4+5)) = 30, where A = 6 obtained by Heron's formula A =sqrt(s*(s-a)*(s-b)*(s-c))= sqrt((6*(6-3)*(6-4)*(6-5)) = 6, and where s=6 is the semiperimeter.
The sides of the excentral triangle are:
a' = 3*csc(1/2*arcsin(3/5)) = 9.48683298...
b' = 4*csc(1/2*arcsin(4/5)) = 8.94427191...
c' = 5*sqrt(2) = 7.07106781...

Mathematica

nn = 500; lst = {}; Do[s = (a + b + c)/2; If[IntegerQ[s], area2 = s (s - a) (s - b) (s - c); If[0 < area2 && IntegerQ[4*Sqrt[area2]*a*b*c/((a + b - c)*(a - b + c)*(-a + b + c))], AppendTo[lst, 4*Sqrt[area2]*a*b*c/((a + b - c)*(a - b + c)*(-a + b + c))]]], {a, nn}, {b, a}, {c, b}]; Union[lst]

Crossrefs

Cf. A188158, A227879.

Sequence in context: A365965 A307130 A379901 * A075285 A361668 A351540

Adjacent sequences: A228094 A228095 A228096 * A228098 A228099 A228100

Keyword

nonn

Author

Michel Lagneau, Oct 26 2013

Status

approved